INCI-mvo‘d HOPF'S BIFURCATION FOR NON-LINEAR FUNCTIONAL , DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO EPIDEMIC MODELS Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY DAVID GREEN, JR. 1976 IIIIIIIIIIIIIJIIIIIIIIIIIIIIIIIIIIIIIIIII ’ ABSTRACT HOPF'S BIFURCATION FOR NON-LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO EPIDEMIC MODELS BY David Green, Jr. .In this thesis we consider systems of functional differential equations with several parameters. Assuming a generic condition on the rate of change of the real part of eigenvalues of the linearized problem, we are able to show the existence of the bifurcating oscillations for the system of equations. The general theorem is then applied to the equations of epidemics to obtain the existence of bifurcating solutions. HOPF'S BIFURCATION FOR NON-LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO EPIDEMIC MODELS BY David Green, Jr. A DISSERTATION Submitted to Michigan State university in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1976 In Memory of My Mbther and Father Mr. David Green, Sr. and Mrs. Rose Bud Green ii ACKNOWLEDGMENTS I would like to thank my major Professors Dr. S.N. Chow and Dr. M.J. Winter for their help and encouragement in the writing of this dissertation. I would also like to thank all of those professors who helped me through— out my academic years here at Michigan State University. I am also grateful for the understanding of my wife Evelyn throughout the writing of this dissertation. I am also grateful to Mrs. Mary Reynolds for typing this thesis. iii TABLE OF CONTENTS CHAPTER I. INTRODUCTION.. Section 1: Infectious Disease Model for Gonorrhea....... Section 2: An Economic Interpretation.. ......... Section 3: Infectious Disease Mbdel..... ........ CHAPTER II. A BIFURCATION THEOREM FOR FDE'S .......... Section 1: Preliminaries. .................... ... Section 2: Space Decomposition. .......... . ...... Section 3: Linear Autonomous FDE's with Real Parameter. ..... . ..... ................ Section 4: Non-linear Autonomous FDE's with Real Parameter. ...... . ........... .... Section 5: The Bifurcation Theorem..... ......... CIIAPI‘ER III. EWPLES... ...... 00...... ........... 0... Section 1: Infectious Disease Mbdel. ............ Section 2: The Gonorrhea Model.. ..... ... ........ BIBLIOGRAPHY.............. iv 13 13 15 19 21 26 4O 4O 55 59 LIST OF TABLES TABLE 3.1. ............................... . ............ 45 TABLE 3.2 ................................ . ............ 55 TABLE 3.3.... ......................................... 58 FIGURE FIGURE FIGURE FIGURE FIGURE FIGURE FIGURE LIST OF FIGURES vi 48 49 SO 51 52 53 CHAPTER I INTRODUCTION The spread of communicable diseases through society involves many disease related factors such as incubation period, susceptibility, infectious period, mode of trans- mission, and resistance to the disease. Also social, cultural, economic, and geographic factors may be considered in any model describing the spread of the disease. To analyze communicable diseases, differential equation models which incorporate some of these factors are useful. In this thesis. deterministic models are employed to study the spread of a communicable disease through society. We use deterministic models rather than stochastic models so that differential equations are used to describe changes in the pOpulation. Stochastic models are necessary when dealing with diseased p0pulation which have very few individuals sick at various times, but these models are very difficult to analyze since there is no single solution x:[t .t1] 4 R, where x(t) is the number of individuals in the p0pulation at time t. Both deterministic and stochastic models are described in the book by N.T. Bailey [1]. Deterministic threshold models are considered in the monograph by P. waltman [11]. These models assume that a susceptible individual does not'become infectious upon first exposure to an infectious individual, but only after repeated exposure to infectious individuals has broken down the sus- ceptible individual's resistance. The population or community under consideration in these models is divided into four disjoint classes which change with time t. The susceptible class, S(t), con- sists of those individuals who can incur the disease but who are not yet infected. Infectious individuals will be referred to as infectives. The infective class, I(t), consists of those who are transmitting the disease to others. The removed class. R(t), consists of those individuals who are removed from the susceptible infective interaction by recovery. The exposed class, R(t). consists of those individuals exposed to the infection, who will as a result become infectious (but are not yet infectious). It is customary to scale the functions I(t). S(t), R(t) and R(t) so that they represent the corresponding fractions of the total populations: I(t) + S(t) + S(t) + R(t) = 1. Deterministic models treat each of these functions as being continuously varying. In the sections that follow we present some communicable disease models and give explicit equations that govern the spread of the disease through society. An analysis of the equations derived from these models is given in Chapter III. §l. Infectious Disease MOdel for Gonorrhea In this section we present an infectious disease model for gonorrhea. The model presented here is by Cooke and YOrke [3]. Individuals who become infected with gonorrhea recover only after drug therapy and do not develop any observable resistance to the disease. After recovery, they immediately become susceptible. The incubation period is from 3 to 7 days and can be ignored when lodking for long term oscilla- tions. we therefore assume that an exposed individual immediately becomes infectious so that R(t) e 0. Also we assume that there is no immunity from the disease so that R(t) a O. The pOpulation for the gonorrhea model is composed of two types of individuals, the susceptibles and the in- fectives. we call these the active population. we assume that the active papulation remains constant. Let x(t) denote the size of the infectious pOpulation and S(t) the number of susceptibles. The rate of new infection depends only on contacts between susceptibles and infectious indivi- duals. we assume that there is a small time lag o, O < o < 1, between contacts with a susceptible and an infective, before new infectives are observed. Since S(t) equals the constant total minus x(t), this rate in effect depends only on x(t) and can be written g(x(t - 0)) for some continuous function g. This model also assumes that there is a single infectious period L, 0 < o < L (the time it takes an individual to seek out and receive treat- ment). Therefore pe0ple are infected at the rate g(x(t - 0)) and are cured at the rate at which they contacted the disease L time units ago, g(x(t - L)). The function x(t) then satisfies the differential equation (1.1) S—E=g 0, this is inter- preted as meaning that there is a constant suprpulation of size c of incurable infectious carriers of the disease. The active pOpulation in the case of gonorrhea is com- posed of two subp0pulations, the infected males and the infected females. This decomposition into suprpulations is necessary for a more accurate and detail analysis in studying the spread of the disease. For males, gonorrhea is easily detected since pain usually develops a couple days after initial infection, whereas with females infection can go undetected for longer periods of time. Thus females can be infectious and is able to transmit the disease without knowing they have it. To study the disease when both the female and male population are considered as separate sub— populations of the active pOpulation, we assume that the number of males with the disease is directly proportional to the number of females with the disease. we call the female pOpulation the main reservior for spreading the disease. This assumption seems quite reasonable when one considers the social behavior of society at large. In this model, x(t) is the size of the total pOpulation. Let cm and cf be the prOportions of the population which are male and female respectively. Let Pm(s) and Pf(s) be respectively the fraction of the infected male and female population which takes longer than time s to be cured after infection begins. Thus Pm(0) = Pf(0) = 1. Let L be the maximum cure time. Choose L large enough so that Pm(L) = Pf(L) = 0. Let P(s) = cum(s) + cfPf(s), then the function x(t) satisfies (1.5). we will delay analyzing the solutions of the gonorrhea model and the following models until Chapter III. §2. An Economic Interpretation we now consider an economic interpretation of the second model for gonorrhea presented above. Let x(t) denote the value of a capital stock at time t. Assume that the rate of production of new capital de- pends only on x(t). and that this rate is given by g(x(t)) for some continuous function 9. we assume equipment depreciates over a time L to value 0. L is the lifetime of the equipment. we further assume that the depreciation is independent of the type of equipment and at time ”a" after production, the value of a unit of capital equipment has decreased in value to P(a) times its original value (so P(O) = l and P(L) = 0). Thus at any time t, x(t) equals the sum of the capital product over the period [t — L,t] plus some constant c, where c denotes the value of non-depreciating assets. The function x(t) satisfies (2.1) x(t) = J‘L P(a)g(x(t - a))da + c o Letting s = t — a, ‘we Obtain (2.2) x(t) = I: L P(t - s)g(x(s))ds + c Equation (2.2) is the same as (1.5) with o = 0. It is easy to see that (1.3) is the differentiated form of (2.2) where p(a) =«ég P(a) is the rate of decrease in the value at age "a" for a capital unit whose value at the beginning of production is l. §3. Infectious Disease Model In this section we present an epidemic model by Heppensteadt and waltman [7]. This particular model is a generalization of the model of Cooke and Yerke [3] presented in section 1. At time t = 0, I0 infectious individuals, called infectives, are introduced in a homogeneous way into a pOpulation of S susceptibles. At any time t >)O, the O pOpulation is divided into four disjoint subpopulations, S(t), I(t), R(t) and E(t). These functions have been defined previously. we assume the disease spreads through the pOpulation according to the following rules: I i) The rate of exposure of susceptibles to infectives at time t is —r(t)I(t)S(t), where r is a positive continuous function. ii) An individual exposed at time T becomes infective at time t if I: [p1(x) + p2(x)I(x)]dx = m 'where p1(x), 92(x) are given non-negative continuous functions and m is a non-negative constant. iii) An individual infected at time t recovers and becomes immune at time t + U, o a positive constant. iv) An individual first immune at time t becomes susceptible at time t + w, w a positive constant. v) The pOpulation remains constant. Let Io(t), -o'g_t g 0 be the function which describes the past history of the infectives. Io(t) is monotone and satisfies 10(0) = I Io(-G) = 0. Assuming that condition 00 (iii) applies to these initial infectives, the future of these infectives is then known. we take this into account by defining an extension of Io(t) to the real line by 0, lt|.2 o 10(t) = 10(t), «53th 10(0) - Io(t - 0). °.S tIg a we also assume the existence of to < 0 such that t foo [pl(x) + p2(x)Io(x)]dx = m. This condition expresses the fact that some of the initial susceptibles must'become infectious before time 0. Any monotone function Io(t) whose extension to the reals satisfies the above integral-equation is called an admissible function. We can describe the spread of the infection by three functional equations in the unknowns T, S, and 1. Equations for R(t) and E(t) can be Obtained from these. The equations for the model will be derived from the rate at 10 which individuals are leaving the susceptible class. To do this, it is necessary to account for those initially in— fective individuals who are in classes S(t) and R(t) for the first time. These will be denoted by Il(t) and 12(t) respectively. The earliest possible entry for an initial infective into class S(t) is t = w, and for t > w. the number of initially infective individuals who are in class S(t) for the first time is the number who recover before t — w. Therefore. '0: t S U.) 11(t) = 10(0) - 10(1: — w). u) g t For 0 S t g u), those initially infective individuals who are now in class R(t) are those who were initially infective at time t = 0 minus those who are still infective from this initial infective population. For t.2 w. those initially infective who are now in class R(t) are those who are in class R(t) for 0 S t g (1) minus those in class S(t) for t.2 w. Therefore, 10(0) - 10(t). Ogtgw 12(t) = Io(t - w) - IOIt). (us t The equations for 1,8 and T that describe the spread of the disease are as follows: t . (3.1) Int) [p1(x) + p2(x)I(x)dx = m, 1'(t) :.-= 0 if t g to. The equation T(t) 5 0 is just a convenience fer eliminating special cases in the equations that follow. Also, to be 11 infective at time t, a susceptible must be exposed before T(t). Therefore, we require that the susceptible population at time t consists of all those individuals who have not been exposed in the interval (7(t - o - m),t). Any indivi- dual exposed in this interval is eliminated from class S(t): exposure before T(t - o - w) results in infection befOre time t - 0 - w. recovery before t - w and readmission to class S(t) before time t. It follows that (3.2) S(t) = 11m + 30 - ft r(x)I(x)S(x)dx 1’(t-O’-w) In a similar manner, the class of infectives at time t consists of the initial infectives who are still infective at time t plus those individuals who were exposed between T(t - o) and T(t). Those who were exposed before T(t - 0) have recovered by time t and those exposed after T(t) are not yet infective. Therefore, we have T(t) T(t-O) (3.3) I(t) = 10(t) + I r(x)I(x)S(x)dx Equations (3.1)-(3.3) constitute the basic equations that describe the spread of the disease through society. The questions of the existence, uniqueness, and continuous dependence of solutions for these models have been resolved [3,7]. Also all the models presented above are very similar in that the oscillatory behavior of their solutions is similar. Each of these models has solutions which eXhibit behavior that indicate the solutions may be periodic. The 12 numerical work of J. Mosevich [9], and the work of Greenberg [4] indicates that the solutions of the functional differential equations which arise in the model by Hoppensteadt and waltman are periodic for certain values of w- In Chapter II, we will develop some general theory for the existence of nonzero periodic solutions to functional differential equations with a real parameter y. Using this theory we show in Chapter III that these models pre- sented above all have nonzero periodic solutions. CHAPTER II A BIFURCATION THEOREM FOR NONLINEAR FDE'S §l. Preliminaries Let En denote the real or complex Euclidean n— space. For r >IO, let G = C[-r,0] be the space of continuous functions from [-r,0] to En with the usual supremum norm. If x is a continuous function defined on [o — r,o + 1).), o e R, A > o, with values in En, let x t 6 [0,0 + A) be the element in C[-r,0] defined by t! (1-1) Xt(6) = x(t + 6). -r.S 6,S,0- we will denote by BV[-r,0], the space of vector- valued functions, n, on [-r,0] which are of bounded variation and are normalized such that n(e)=o ego E(B) = n(-r) e g_-r and n is left-continuous on [-r,o). Consider the autonomous linear functional differential equation (1.2) x(t) = L(xt) 13 14 where L(-) is a continuous linear Operator mapping C[-r,0] into En. By the Reisz representation theorem, there exists a unique n x n normalized matrix n(:) e BV[—r,0] such that (1.3) L(cp) = 1" [dn(6)]cp(e). e e C[-r,0]. -r Let x(m). be the unique solution of (1.2) through (O,¢) and let T(t) be the bounded linear operator mapping C[-r,0] into C[-r,0] defined by (1.4) T(t)cp = Xt(cp) It is well-known that {T(t):t 2;0} is a strongly continuous semi—group of bounded linear Operators on C[-r,0] (see Hale [5, p.94]). The infinitesimal generator A of the semi-group T(t), t.2 O is given by {p(a) -r g e < o L(cp) = j‘° [dn(6)]cp(9). e = o -r (1.5) Am(9) = It is known that the domain of A consists of all continuously differentiable functions (p(a). -r g e g o, with (1.6) 1im M = [dn(e)]op(e) ago. e .. o I: It is welleknown that if A is defined by (1.5), then the spectrum of A, 0(A) is the same as the point spectrum of A and possibly A = 0. .Moreover, A e 0(A) if and only if (1.7) det Am = det[u - 1‘0 e"6 dn(e)] = o —r 15 where A(x) = XI - f0 e)‘6 dn(6)- Equation (1.7) is called -r the characteristic equation of (1.2) and its roots are the eigenvalues of (1.2). For any real B, the set [xzdet AU.) = 0, Re ). > B} is finite. Given a fixed eigenvalue No of (1.2), the null spaces n(A — x01)J of (A - IOIIJ, j = 1,2,..., satisfy the following nested relations: n(A - 1.01) c n(A - on)2 c: -- - . There exists a smallest integer k.2 1 such that n(A - x01)m is a subspace of n(A - ADI)k for all 1 g m 3k, and n(A - L01)m = n(A - lonk for all m 2 k. The generalized eigenspace Px0 ‘belonging to A0 is defined )0 is the complementary subspace of P , the following prOperties are known: ’0 i) dim P10 < e, where dim Px0 = algebraic multiplicity of to . "o _ k k — R(A - A01) , the range of (A - X01) . tobe 11(A- ADDk. If Q ii) P and Q "o ’o iv) C = P O Q). , where 9 denotes direct sum. ’o o §2. Space Decomposition are invariant under T(t). iii) Q In this section we consider the projection “P on the space C = C[—r,0]. Definition 2.1. An eigenvalue of (1.2) is called simple if its algebraic multiplicity is one. 16 Theorem 2.1. Let ID = ivo, v0 51 0 be a purely imaginary simple eigenvalue of (1.2) and let PAD be the generalized eigenspace belonging to go with projection 11 :C[-r,0] -o P , PM *0 then (2.1) HP (cp) = mo<¢omp> for any cp E C[-r,0], ()0 e C[O,r]. )0 x09 (2.2) (90(9) = e a0, -r g e g 0 T " s '1' , -1 (2.3) (0(3) = boe x0 (bob ”oIao) , 0g 3 g r and <-, -> is a bilinear form defined by (2.4) <).op> = (25mm) - f° I: (T(g - eIaneIcp(§)d§. -r where a0 is the unique solution of A(>O)a = O with Ial = 1, 1.3 [bl = 1. ("T" denotes the transpose). is the unique solution of bTAuO) = O with Proof: Clearly cpo 6 n(A - x01) Since qu0 = xocpo implies (A - ‘01)“)0 = 0. Thus by definition of a simple eigenvalue P>0 consists of exactly those scalar multiples of (po. This shows the uniqueness of a Similarly we have 0. . 'r * _ 'r “"os the uniqueness of b0. Let ¢o(s) - b() e , 0 g s g r, then 17 (6-5) A 5 'r b0[I - fir]: e"o d'n(6)e ° d§]ao )Oe = by: - I: dn(e)ee 1ao * T . bOA (>.o)a0 where A e A’(>b) = I - J'0 dn(e)ee ° . -r It follows from the proof of Lemma 21.2 in [5, p.109] * , , * that <¢O.mb>)#'0. Therefore ¢0(8) is defined. #'0 also follows from more general results in functional analysis (see, for example [10]). - s <¢o.qo> = (bge x0 (bgA'(ko)ao)-1.qo> T . -1 'r “‘08 = (bgA '( x0) a0) -1 <¢;.ch> (bga'uomofl (bgA’UDHOI = 1. Therefore, HP (CD) = ¢O¢ In applications, system (1.2) is often given in the real Euclidean n-space RP. For this reason, it is often desirable to Obtain the projections and the constant of . . . n variation formula in R . 18 Suppose (1.2) is real. All complex eigenvalues of (1.2) appear in complex conjugate pairs. Assume )0 = 1Y0 is a simple eigenvalue of (1.2). Hence 30 = -ivo is also an eigenvalue of (1.2). Let P be the generalized eigen— space belonging to [ivo,-iuo]. Then, dim P = 2 and Q = (Re m0, im m0) is a basis for P (where wb is as O in Theorem 2.1). Theorem 2.2. Let IO = iVo' V0 >»O be a simple eigenvalue of the real system (1.2) and P be the generalized eigenspace belonging to {ivo,-iyo}. Then the projection HP onto P is given by (2.5) flp(m) = §o where N39 (2-6) mo(6) = e 60. -r.S 9.§ 0 'r 'r . -1 " s (2.7) ¢0(s) = bo(boA ()0)ao) e to , o g s g r where 90 = (Re %0 Im CPO) (2.8) Re Y0 =( I0) Im *0 a0 6 En is the unique solution of A(Ao)a = 0 with [a] = 1, b3 is the unique solution of bgA(xo) = O with 1b] = 1. 19 Proof: 60 = (Re $0.1m mb) is a ba31s fer P. By Lemma 21.4, Hale [5, p.109], we have ((0430) = l, = O, (Wo,qo> = O and = 1. These imply that <20,§0> = 1 O). (O 1 This proves the theorem. Corollary 2.1. For any m E C[-r,0], (2.9) HP(cp) = (Re I{lopcp>Re mo + (1111 ¢O,cp>Im cpo Proof: Since (Re $0,Re mo) = 1, (Re $0,1m.q0> = 0, (Im ¢O,Im $0) = l and (Im wo,Re go) = 0, writing out (2.5) gives the corollary. §3. Linear Autonomous FDE's With Real Parameter In this section we consider (1.2) with a real parameter y. ‘With real parameter y, system (1.2) becomes (3.1) E(t) = L(y,xt) where for each y, L(y,-) is a continuous linear operator mapping C[—r,0] into B“. we assume that there exists a unique n x n normalized matrix n(y,-) 6 BV[-r,0] such that (3.2) L(y.¢) = j° -dn(y.e)]m(6). e e C[-r,0]. -r and the dependence of n on y is smooth. By smoothness, we mean the following: (3.3) n(y.e) = no(e) + yn1(e) + Y2n2(9) + o<|v|2) 20 where no, n1 and n2 6 BV[-r,0]. The characteristic equation of (3.1) has eigenvalues which also depend on y. Let y = be fixed. If Re A < O for every eigen- ”’0 value x of (3.1) with y = VO' then the zero solution of (3.1) for y = V0 is exponentially stable. Suppose for a different value of y, say y = Y1: there exists a pair of complex conjugate eigenvalues of (3.1) with real part greater than zero, then the zero solution of (3.1) for y = Y1 is unstable. we assume that there exist such a and V1“ Re A *0 is a continuous function of the real parameter y and therefore by the intermediate value property for continuous functions there is a y, say y = l; with Y0 < Q, < Y1 and a x(?) such that Re I(?) = 0. ‘Without loss of generality we may assume ?'= 0. we further assume that the eigenvalue, x(y) = g(y) + iv(y), of the characteristic equation of (3.1) satisfies near y = 0, no + up! + uzvz + 0~ by (4.6) <).cp> = I((OIOIOI - 1° (3 ((s - 9)[dn(6)]cp(6)ds -r we may assume that )= I, where I is the identity matrix. Also if the decomposition of any element m in C is written m = ¢P + m9 where mp is in P and To is in Q: then mp Q(v). ‘we now consider the following situation. Let y be fixed, *0 = p + iv and i0 = u - iv be simple eigenvalues of (3.1). Let P be the generalized eigenspace associated with 10 = p + iv and 10 = H - iv. Then the dimension of P is 2, and the Operator A defined by (1.5) satisfies AP CLP. Therefore there is a 2 x 2 matrix B such that A4 = 4TB. where 9 is the basis for the generalized eigenspace P. The subspace P is spanned by eigenfunctions of the form eke, -r‘g e S_O. we will now determine the form of the matrix B. Let ¢b be the eigenfunction corresponding to yo. Since the system is real, Eb is the eigenfunction corres- _ 6 ponding to A0. .Moreover, qo(e) = ex0 a0, a0 6 En. Let 41(9) = §Iopo(e) + 30(9)] 42(9) Eli-[40(9) - 60(9)] 24 The eigenfunctions $1 and $2 are real and are a basis for P. we will consider this basis because the Banach space C = C[—r,0] is real. Applying the operator A defined by (1.5) to these basis elements, we have 1 _ 5 EIISOIGI + 90(9)] (4.7) Am1(9) = §INn¢oIBI + ioab‘e’] = uml - 442(9) 1 d - figquoOIe) - 430(9)] (4.8) Am2(9) = {Inflow - 1060(9)] = (142(9) + vcpfle) = VCp1(9) + “(92(9) Therefore, (4.9) A = (41(9).e2(e))(_3 :) The matrix B is given by _ u v (4.10) B _ (w (1) Thus the matrix B is strictly determined by the basis for * the generalized eigenspace P. The Operator A defined by (4.4) has eigenfunctions of the form e'”, o g e g r. We 'will now prove the following. Theorem 4.1. Let y ‘be fixed. Let I(Y) = p(y) + iv(Y) and 1(y) = p(y) - iv(y) 'be simple eigenvalues of (3.1), and P be the generalized eigenspace associated with x(y) and I(v) ‘with projection 25 HP:C[-r,0] 4 P Consider the real non-homogeneous system (4.1) . If X(Y,Qp) is the unique solution of (4.1) through (0,¢) and w(y) (t) = then (4.11) fi-ww) (t) = B(Y)W('v)(t) + Yo(v)(O)N(Y.Xt(y)) (4.12) zt = HQT(Y) (t - (mp + no]: T(y) (t - s)on(y.xS(y))ds where no = 1 - HP and 2t = [Ith MY) vhf) (4.13) B(y) = -v(YI MY) 2521:: Let x(y) (t) be a solution of (4.1), and y(y) (t) be a solution of the adjoint equation to (4.1) on (-.,o]. For each t e (-.,O], let y(y)t be the element in C* = C[O,r] defined by y(y)t(a) = y(y) (t + a.) for a e [0,r]. Then by Theorem 17.1, [5, p.90] we have for all t 2 o, t t 0 =f Y”) (8)N(Y.xs(y))ds + (YIY) .xo(v)>- a Each row y the matrix e-B(Y)e. YOIY) (O), 0 S e g s is a solution of the adjoint equation on (-¢, a) . Therefore 42'3”” Y0(y)xt(v)> = j: e’BI‘I’s (ow) (O)N(v.xs(y))ds + 26 t eB(y)(t-s) (YOIY).Xt(Y)> = I YO(Y)(0)N(y,xS(y))ds 0' + - Let (My) (t) = then My) (0) = (YO(Y).cp(Y)> Therefore. w(v)(t) = (t eB‘YIIt'S’ (0(y)(o)N(y.xs(y))ds O + eBW) (t-O) w(y)(0) Differentiating with respect to t, ‘we get 5; B(v)(t-o) dt w(v)(t) = B(Y)e (MY) (0)N(v.xt(y)) Let 0 = t, then 391? (My) (t) = B(y)m(y) (t) + (OH) (O)N(Y:Xt(Y)) Equation (4.10) follows immediately from (4.2) be taking projection IIQ on both sides. This completes the proof of the theorem. §5. The Bifurcation Theorem In this section the main theorem is proved. we shall show the existence of non-zero periodic solutions using a technique employed by Hopf for ordinary differential equations with a real parameter y [2,6]. 27 Theorem 5.1. Consider the non-linear system (5.1) x(t) = L(y,xt) + N(y,xt) where y is a real parameter, L(y,-) is a continuous linear Operator mapping C[-r,0] {into En, and the associated linear system (5.2) {<(t) = L(y.xt) Assume i) N is Fréchet differentiable and N = 0(Iml) uniformly on bounded sets of y. ii) For y = 0, there exists a unique pair of simple purely imaginary eigenvalues ivo, -ivo. v0 # 0 and no other purely imaginary roots that are integral multiples of ivo. iii) Re x’(o) 7! 0. Then there exist non-zero periodic solutions bifurcating from y = 0. To prove Theorem 5.1, we need the following lemmas. Let go and $0 be the eigenfunctions corresponding to No and 10' It is known that (see Section 2) A09 ¢b(6) = e a0, -r g 9‘s 0 -A 8 (0(3) = bge ° (bga'uonorl. o g s g r with 28 = 10 <$OI (p0) = 0 ll H <‘Io'°-'30> = 0' @0598 Let 691(9) = %[coo(9) 4' 50(9)] 42(9) = Eli-[40(9) - 80(9)] (1(9) = §Iwo(e) + 30(9)] (2(9) = Eli-HOW) - 170(9)] T(Y.t) = T(th). T(O,t) = T(t) we will now prove the fellowing lemma. Lemma 5.1. Let T(t) = T(O,t) and gi(t) = (Yi,T(t)¢1>. Then 331. o at 332. 39. at t=“b 2 Proof: Applying the solution operator T(t) to “0 and Eb we have T(t)cpo = e cpo T(t)§b = e ob 29 Therefore 1 _ .. = 4<¢0 + WO.T(t)(¢b + moi) Act 1 - Int 1 - = z’+ Z<¢Ooe go) It follows that =——lt_qo t " t = Ill; AO + % AO] =°‘b %(x0 + 10) = 0. Also (Y T(t) = 1<$() - I ) T(t)( + " )> 2' Op1 410 0' OF’o CI’o t 't 11 11.. .. = 4 Differentiating with respect to t, we Obtain .53E1 —§<: T(t) >| 41% — I at t-ub at 2' ‘91 t-wo 4i ‘0 "o _ 4i VI0 2 This proves Lemma 5.1. Lemma 5.2. Let T(y,t)cqD = xt(y) where 9b is complex and x satisfies (3.1). Let t V(t) = any XI?) (t) then 3O ) ) Rot . L (v + L (m e (5.3) v(t) = <0 t 1 0 v0 = 0 where A0 is a complex eigenvalue of the characteristic equation of (3.1) and (5.4) I Lo(cp) = jo [drb(e)]cp(e) -r (5.5) L1I¢)I= Ifr [dn1(6)]m(e)o Proof: Since x(0) = ¢(O) and .x(y)(e) = p(a) for — —.a_ = __B_ = all e e [-r.0]. then v(9) - BY XIYHB) BY (p(a) 0 for all e in [-r,0]. Therefore v0 = 0. we also have i(y)(t) = L(v.xt) = 1" tan”. 9) ]X(y) (s + e) -r Therefore x(th) = X(v)(0) + jtj" (dim. e)]x(v) (s + ems O -r Since the initial conditions for any solution are the same, ‘we have x(v) (t) - x(o) (t) = jg]: [dn(v.e)]x(v) (s + ems - I: Lo(xs(0))ds .4 I21: [dno(9) + vdnlm) + 0(IvI2)]x(y)(s + e)ds — j: LbIXs(O))ds = j: LO(XS(Y) - xs(0))ds + v]: L1(xs(v))ds + o(lvl2) 31 Therefore Y)(t) - (oHt) _ 1 t x(, y x _ I} I0 Lo(xs(y) - xs(0))ds] + I: L1(xS(Y))ds + 0(Iyl2) lot Letting y 4 O and recalling that xt(0) = mbe , we get 8 v(t) = I: Lo(vs)ds + I: L1(¢b)exo ds. Differentiating with respect to t, 'we get th v(t) = Lo(vt) + L1(¢0)e This proves Lemma 5.2. Lemma 5.3. Let A1 = x'(O). Assume Re l1 # 0. Then '1' . __ 'r ’o >abo‘) ”o’ao " boL1Ie ”‘0‘ Proof: Let x(y) be an eigenvalue of the characteristic equation det A(x(y)) = 0 with corresponding eigenvector a(y). By a result in Hale [5, p.99] we have [MYII - [0 [dnm eHeMY) e1am) = o. -r Differentiating with respect to y at y = 0, we have A09 ]a 9 [x'(0)I - JO [6111(9)]e)‘o - )6ij [dno(e)]ee -r -r e ‘0 ]a‘KO) O = o. + MOI-[o [dno(e)]e r Therefore 32 A 9 A06 0 0 [A11 - i1f_r [drb(OIJBe ]ao - (j: [dnl(e)]e ]a0 6 + [x01 - j: [drb(e)]ex° ]a’(o) = 0 Since b3 annihilates the range of MAO), we have 6 b3[ x01 — fr [dno(e)]e)b ]a’(O) = 0. If we multiply the above . T T . _ T ‘0' expre531on by bO 'we get, xiboA (x0)aO — boLl(e )aO X06 where A’().O) = [0 [d'rb(9)]9e . This completes the proof —r of Lemma 5.3. Lemma 5.4. Let v(t) be as in Lemma 5.2. Then 21r Y(t) = (woovt> By Theorem 4.1 (in the complex form) and Lemma 5.2, . )‘ot y(t) = koy(t) + ¢O(O)L1(cp0)e Since vO E 0. y(0) = 0. Therefore (t—s) s Y(t) = [t e)“ to(0)L1(q)o)e)'o ds 0 A t = e 0 I: ¢O(O)Ll(¢o)ds t o = ex0 It bg(bgA’(xo)ao)-l Ll(e)‘o )a0 ds 0 t Ale)V0 It ds (by Lemma 5.3) O th = Alte . 33 Therefore (WO.V(u)o)> = .35 )1. This follows from the fact that “’o = €15 and x = ’b = ivo at t = ”0‘ This proves Lemma 5.4. Lemma 5.5. %¢1,T(y) (u) o)cpl> = 25-0 Re A1 and 3%«2 um) «howl: = ;6- Im x1. Proof: NHNH _§:{<¢1'T(Y)(w0)¢1>4—§;<¢:+ Eo'T(Y) (m0) $0 + T(Y) (nb) 60> + %<¢o,a %T(v) (wo)cpo> + 4(60' '36; R(y) (ub)> by Lemma 5.2 i<¢o.v(mo)> + fidofimb» l - _2__1T .—.. x —+ ), by Lemma 5.4 4 1 vb 4 1 “O =-'-—-- Re I. 2vo 1 Similarly a _ l “(IT‘NYI (mo)¢1> - 35% 'Io '3‘": TH) (0)0)ch + 35-- T(YHuqub) _. 1 1 - — - ZI - Z§<¢Oov(qo)> .. _1. 21: _ _1. 21 = .23: - 45. X1 v0 45. XI. v0 v0 Im >‘l We will now prove Theorem 5.1. Proof: The existence of urperiodic solutions of (5.1) is equivalent to solving o T(v) (on + (2’ Th!) ((1) - S)X0N(v.xs(v.cp))ds T(wOICp + [T(Y) (wIcp - T(ub)cp + I: T(v) (u) - s)XoN(Y.Xs(y.cp)]ds. Let P be the generalized eigenspace associated with A = {iVo'-ivb} and Q be the complementary subspace of P such that C = P GIQ. we scale the above equation by letting T = 6(m1 + I) where 6 ‘belongs to R, I 6.0, and ¢1 E P. Let HP and no be the projections of C[-r,0] onto P and Q respectively. The projection n on the scaled P equation for cp(q)P = 11P(cp)) gives nP{T(v)(m)(eo1 + 64) - T(qDIIeol + 6)) + I: T”) (u) " SIXONIYoxs(y.€cpl + ands} = o where T(mb)¢P = mp. Let '3 e C[-r.o]. By Theorem 2.2, Rpm = <¢I,¢>¢1 + (#2.$>¢2 ‘where T1 and $2 span the subspace P and (1. t2 belong to the span of the generalized eigenspace P* of the adjoint equation of (5.1) associated 35 with A. Therefore UPS = 0 if and only if (Il'$> = O and <¢Z,$> = 0. Thus we have + if: T(v) (u) - sIXON(v.xs = 0 (W20T(Y)(w)(cp1 + In " T(ub) ($1 + W) + 46-1”: T(v) (u) - smouwncsaw.(an1 + e())ds> = 0 Because there are no other eigenvalues that are integral multiples of ivo, the operator (I - T(qo)) is invertible. Thus the projection HQ(¢Q = nQ(¢)) defined by (4.12), on the scaled equation for T gives, )- (I - T(won'lInQ'uy) ((1)) (el + I) - T(ub) (cpl + I) +-1é nojg'rmm - s)onw.x,=.,(ecp1 + e)))ds} = 0 Let G be the map from R3 x Q into R2 x Q defined by G = G(€,Y.w.¢) = (Gl,Gz,G3) where G1 = ()1.T(y)(w)(o1 + t) - T(ub)(01 + I) + éfw T(Y)(w - SIXbN(vas(Yo€¢1 + GIIIdS> O C) II 2 <)2.T(y)(m)(o1 + 4) - T(mo) (el + (I) + if: T(y) (u) - s)X0N(y,xs(y,ECp1 + €$Hds> G3 = v- (I - T(mo))—1[HQT(y) (m) (cp1 + I) - T(ub) (o1 + I) + % 110]: TM) ((1) - s)xON(v.xs(v.ecpl + ends}- 36 Note that G is well defined for 6 = 0 since N = 0(I¢I). Also G is continuously differentiable in a neighborhood of (0,0,wb,o) since each 61' i = 1,2,3 is continuously differentiable. By Theorem 2.2, the zeros of G are solutions of (5.1). Let 6 = y = I = 0. Since T(O)(mo) = T(mb) and N = 0(ImIIo we have G1 = Gl(o,o,mb,o) = 0 and G = G2(O,O,qo,0) = 0. Also for y = O, 2 UQT(0)(mb)(cp1 + w) = T(qo)(m1 + TIQ = I- Therefore for D defined by D = 6G2 362 3G2 aw BY 3) 6G3 363 363 LOU.) BY BI) .4 (E. Yul): U) = (oooouboo) To prove that D is an isomorphism from R2 x Q onto R2 x Q, it suffices to show that the matrix A defined by r- q 361 3G1 6w BY A = 562 3G2 _30) BY d (60 Yo I”: w) = (0000 “b00) is non-singular and that 3G3 .3;— (60 You): U) = (0,0.ub,0) 37 is an isomorphism on Q. We compute the entries of A as follows: Therefore, 6G ...; _ _8_ OLD (0:00 flbIO) — Bu.) (wl'T(w)ml>lquO = O by Lemma 5.1. Similarly, G2 = 62(0000 Woo) = ($20T(w)cp1 " T(Ub)m1> and 3G 2 .. _EL SID—- (000: “1000) — Bu) (WZOT(w)m1>IUFWO V0 = _2— by Lemma 5.]. Computing the partial derivative of G1 with respect to y, we Obtain 5G1 _ .1 ST (o.y.mo.0) “ <‘Il' av T”) (WW) <(1. {in x(v.o1)(mo)> 17' 2V0 Re II by Lemma 5.5. 38 Similarly, Therefore 6G2 a 57" (0,v,u)o.0) - <‘I2' 5? '1‘”) (wow)? = <42. 35:, X‘Yocpl) (ub)> v 2V Im A1 by Lemma 5.5 0 Thus the matrix A is given by - w 0 -—— Re A 2V0 1 A = V 0 w -— --- Im A 2 Zvo 1 Since Re k1 # O, A is clearly non-singular. we now com- 363 pute ST 0 GB = 63(0100 (lb: ‘1‘) = U 66 Therefore, SE—-= I, where 1:0 4 Q is the identity map. we therefore have that D is a linear isomorphism from R2 x Q onto R2 x Q. Therefore, by the implicit function theorem in Banach spaces ([8, p.17]), there exist C’ maps w(€). v(E) and 4(6) defined for 6 in ('50'60) where ' 2W > 0 such that (”(0) = = —-, v(o) = O, ((0) = 0 6. mo .0 and G(€.v(e).m(€).¢(e)) = 0. "£0 < 6 < 60- This gives 39 a one parameter family of non-zero periodic orbits bifurcating from (o,x0). This completes the proof of Theorem 5.1. Definition 5.1. Let I:C x R 4 R where C = C[-r,0]. I is called a first integral of (5.1) x(t) = L(y,xt) + N(y,xt) if, for every solution x I(xt,y) = constant for all ti t > 0. Theorem 5.2. Consider the non-linear FDE (5.1). Assume (5.1) has first integral I, and I:C x R 4 R is smooth. Assume that M,= [m:I(¢,y) = C, C = constant, y E R] is a smooth manifold of co-dimension one in C. Let Q be the smooth vector field defined on M induced by the solution of (5.1), and 5 e.M. be a critical point of 4. By using local coordinates around g, *we obtain a FDE in the form of (5.1). Then the conclusions of Theorem 5.1 is valid for the vector field Q. CHAPTER III EXAMPLE S §l. Infectious Disease Model As an application of the theory presented in Chapter II, we consider the infectious disease model of Hoppensteadt and waltman [7] given in Chapter I. In this model the spread of the disease is governed by three functional differential equations in the unknowns T. I and S. These equations are: (1.1) [2) [91(x) + 92(x)1(x)]dx = 111. T(t) E 0. t S. to T t (1.2) S(t) = 11(t) + 50 - jt r(x)S(x)I(x)dx T(t-G-w) (1.3) I(t) = Io(t) + [T(t) r(x)S(x)I(x)dx T(t-O) The functions 11 and ID are as defined in Chapter I: o. |t| _>_ o 10(t) = 10(t). -GSt_<.° 10(0) -Io(t-o). ogtgo 0: t S (D 11(t) = 10(0) - IOIt - w). T‘s t 40 41 The function I1 gives the fraction of those initially infected at time t = 0 who are now in class S(t) for the first time, while for t E [-o,o], Io(t) describes the past history of the disease. The function Io(t) has been extended to all reals so that the future behavior of the initial infectives is known. The value to in (1.1) satisfies, t0 ]‘ [p (x) + p (x)I (x)]dx = m. 0 1 2 O In our study we are interested in the behavior of I and S for large time t, so without loss of generality we consider the differentiated forms of (1.2) and (1.3) for t.2 o + m. (o is the fixed time during which an infective remains infected, and w is the period of immunity). For ti; o + m. the functions Io(t), 16(t), and I£(t) are all zero ("’" denotes derivative with respect to time). The differentiated forms are: (taking T(t)==t) (1.4) s’(t) -r(t){I(t)S(t) — I(t - o - m)S(t — o - (0)} (1.5) I’(t) r(t){I(t)s(t) - I(t - o)s(t - 0)} Every pair of constants is a solution of (1.4) and (1.5). It is customary in the stability theory of non- linear differential equations to study solutions which are "close" to constant solutions. To study the stability of (1.4) and (1.5) we choose a particular pair of constants 42 a, B which solves (1.4) and (1.5) and linearize about this pair of constants. we make the following substitutions (1.6) I(t) Y(t) + I3 (1.7) S(t) X(t) + (1 Substituting (1.6) and (1.7) into (1.4) and (1.5) respectively and for simplicity choosing r(t) = c, c is a constant, we Obtain, after drOpping non-linear terms, (1.8) km = -cB{x(t) - x(t — o - wH — ca[y(t) - y(t - O - w)] (1.9) {r(t) = cB[x(t) - x(t - 0)) + ca{y(t) - y(t - 0)) Note that (1.8) and (1.9) are functional differential equations with real parameters c, 8, d, O and To In Chapter II we develOped some general theory for non-linear functional differential equations with real parameter y. we showed that if A 'was a simple complex eigenvalue of the non-linear functional differential equation (II, 4.1) ‘with dependence on the real parameter y, such that Re A'(v = o) #'0, then (11.4.1), has non—zero periodic solutions. we showed such existence in part by studying the zeros of the characteristic equation associated with (II, 4.1). Therefore to examine the behavior of the solutions of (1.8) and (1.9), we shall study the solutions in terms of the characteristic roots of the characteristic 43 equation associated with (1.8) and (1.9). To obtain the characteristic equations for (1.8) and (1.9) we let x(t) = kle)‘t and y(t) = kzext' where k1 and k2 are constants, k is an eigenvalue of the characteristic equation associated with (1.8) and (1.9). Substituting these values of x(t) and y(t) into (1.8) and (1.9) we Obtain (1.10) xkl = -c(3[k1 — kle'*(°+w)} _ ca[k2 _ kZe—i(0+w)} (1.11) xkz = c8{kl - kle-xo} + ca[k2 - kze-xg} Rearranging equations (1.10) and (1.11) we Obtain, (1.12) {x + c8 - ope-x(o+u”]kl + [co - cue-I(O+w)]k2 = O (1.13) [cBe-xo - cfilk1 + [x — ca + cue-XOIk2 = O The characteristic equation for system (1.12) and (1.13) is Obtained by solving the determinant below: A + CD - cBe-X(G+w) ca - cue-x(o+w) -A0 A0 cBe - CD A - ca + cae- Evaluating the determinant, we obtain (1.14) f(x,a,8,c,m,o) = x2 + CBX _ che'*(°+w) - cox + lcae-xo = 0 From the model, we fix c, o, a, and B, then (1.14) becomes (1.15) f(x,w) = A2 + cfil - che-A(O+w) - cox + lode-IO = 0 44 Note that A = 0 is a solution of (1.15). This expresses the fact that every constant is a solution of (1.8) and (1.9). It is of interest to study the zeros of (1.15) for l # 0. Applying Theorem 5.2 to (1.8) and (1.9) we Obtain the equivalent form of (1.15) for A # 0. This equivalence is given by A0 -A(U+w) - c0 + c0e- = 0. (1.16) f(l.m) = x + CB - CBe we examine the roots of the characteristic equation (1.16) with values of c = 0.2 and 0 = l. we also assume x = iv and x = -iv are purely imaginary eigenvalues of (1.16). With these values of c, 0 and l. (1.16) becomes (1.17) f(iv.w) = iv + 0.25 - 0.20e'i"(1+‘”) — 0.20 + 0.2ae'iV = 0 Equation (1.17) is equivalent to (1.18) iv + 0.28 - 0.28 cos v(l + w) + 0.2i8 sin v(l + w) - 0.20 + 0.20 cos v - 0.2i0 sin v = 0 Equating real and imaginary parts we Obtain (1.19a) 0.28 - 0.28 cos v(l + w) + 0.20 cos v — 0.20 = 0 (l.l9b) v + 0.28 sin v(l + w) - 0.20 sin v = 0 For fixed v and u) we solve (1.19) for a particular pair of constants 0 and 8. Table 3.1 shows values of 0 and 8 Obtained by fixing values of w in (1.19). 45 TABLE 3.1 v w 0 3 v/z 1 7.85398 3.92699 W/2 2 3.92699 3.92699 W 2.25 3.25323 22.21441 W/4 3 5.75094 19.63495 w/4 4 . .56270 4.74030 w/4 5 .81331 4.74030 W/B ‘ 5.23599 4.88524 For each fixed 0 and corresponding 0 in Table 3.1, we plot 8 versus v to find additional values of 8 and v which solve (1.19). There are an infinite number of pairs (v.5) which solve (1.19). For each choice of 0 and w‘ choose that pair (v.8) for which iv and -iv are closest to the origin. Other pairs of intersections in the vB-plane will not be considered. In Figure (3.1), we have 0 = 7.85398 and w = l. The graph is for 8 versus v. There appear to be no intersections. However, there is an intersection at infinity since 5 _ 0 sin v - 5v - sin 2v by (l.l9b). 46 FIGURE 3.1 47 In Figures (3.2)-(3.7), there are several pairs of (0,8) intersections shown. It appears from Figures (3.2)-(3.7) that for each such pair (v,8) which solves (1.8) and (1.9), there are no integral multiples of v such that (kv,8) solves (1.8) and (1.9). At these intersections we will show first that Re A’ # 0, then, by applying Theorem 5.1, that the system (1.8) and (1.9) has non-zero periodic solutions, at least for the values of 0, 0 and w used in each graph, Figure (3.2)-(3.7). TO ShOW d RSBX(fi) # 0' at Bil i = 0,112,000, we differentiate (1.16) with respect to 8 and Obtain (taking 0 = l) (1.20) gg-+ c - c[—8(l + w)e'*(1+w’.%%-+ e‘*(1+w’] -1 91.. — c0e d5 - 0 dA Solv1ng for 554 we have 9-2: — ce-XCL'I'UJ) _ C (1.21) — d5 1 + c8(l + m)e’*(1+w) - c0e -A The 8's which solve system (1.19) in Figures (3.2)- (3.7) correspond to those eigenvalues A which solve (1.16) with real part equal to zero. Thus for all such 8's, A(B) = iv. The pair (vj,8j) such that ivj and -ivj are closest to the origin will be denoted by (vb.80). At 8 = 80, (1.21) becomes 48 mmmNm.m n O D FIGURE 3.2 49 ( mmmmm.m e ( FIGURE 3.3 SO A ¢m0m5.m H FIGURE 3.4 51 7.9 r — — '7 H17 .56270 O. 41:4 :4: ...:4 :4: I I - I'I ‘- o O n ‘I b I b g . 2 ~> J' l ,( FIGURE 3.5 52 H m m .-I (D II C3 c-l ll 0 V' II 3 I...- FIGURE 3.6 mmmmm.m u d 53 FIGURE 3.7 54 92. (1.22) dBIB=Bo .c[cos vo(1+w) -i sin v0(l+w) -1} 1+cfio(1+w) (cos VO(T+ w) -1 Sin vo(1+w)-ca(cos vO—51n v03 writing (1.22) in the form B + hi and considering only the real part, we obtain (1.23) d R§B3151|a=fi0 = {c cos “0(1 + w) - c + c260(1 + m) c260(l + m)cos yo(l + w) cza cos Yo(1 + w) + cza cos “0 2 . . c a Sin vb(l + w)31n yo}/ 2 (1 + cBO(l + w)cos v0(1 + w) - CO COS v0) + (ca sin v0 - caou + w)sin vo(1 + w))2 Computed values of Re x’(Bo) are given in Table 3.2. The calculations show that Re X'(BO) # O for all values of B0 considered. Thus for each pair (Vb’BO) we have Re l(fio) = 0,. Re x’(Bo) #’O and lo = ivb. Remarks. (1) NO is a purely imaginary simple eigenvalue if x0 satisfies, A(iyo) = O and A'(iVo) #'0. These conditions are easy to check and are not considered here. (2) To check that there are no integral multiples of ivb. we note that there are finitely many roots of the characteristic equation (1.16) that are purely imaginary. If there exist integral multiples of iv . then there 0 exists a largest integral multiple niyo. Let v1 = nyo. 55 Applying Theorem 5.1 to v1 ‘we Obtain the desired results. That is (1.8) and (1.9) has non-zero periodic solutions. TABLE 3.2 a B v Re l'(5) w 3.92699 3.92699 1.57080 .428318 2 18.9612 42.5261 9.21942 4.19972 2.25 5.75094 115.682 5.14383 5.26839 2.5 .562698 11.93520 2.77406 .32172 3 .81331 7.08452 2.18788 1.29186 4 5.23599 4.11505 1.74635 1.30180 5 §2. The Gonorrhea Model As a second example, we consider the gonorrhea model prOposed by Codke and Yorke [3]. The model is described in Chapter I. The spread of the disease through society is given by (1.24) in» = g(x(t - on - g(x(t - 1.)) where 0 < o < L. Without loss of generality we choose L = 1 by a change of time scale. The rate of new in- fection is given by g(x(t — 0)); g(x(t - 1)) is the rate at which the infectives are being cured assmming they contacted the disease 1 time unit ago. One choice for the function g is (1.25) [g(u) = au(1 - u), a > 0 constant. 56 This choice of g is reasonable since at u = 0 there are no new infections and at u = 1 all new infectives have been cured. With this choice of g, (1.25) becomes (1.26) R(t) = ax(t — o) - ax(t - 1) - ax2(t — o) + ax2(t - l) Linearizing (1.26) about zero, and considering only the linear part, we Obtain (1.27) R(t) = ax(t - O) - ax(t — l). The solution of the linear equation (1.27) can be studied in terms of its charactersitic roots. To Obtain the characteristic equation. let x(t) = ext ‘where l is an eigenvalue of the characteristic equation of (1.27). Substituting x(t) into (1.27) we Obtain (1.28) l = ae-xd — ae-x Clearly x = 0 is a characteristic root of (1.28). Applying Theorem 5.2. we eliminate the zero solution of (1.28). Assume l = iv is a purely imaginary simple eigenvalue of (1.28), then separating real and imaginary parts of (1.28) we Obtain (1.29a) a cos v0 - a cos v = 0 (1.29b) a sin v - a sin v0 = v. 0 < o < l. 57 The first equation in (1.29) shows that if v0 = -v + 27m, m an integer, then any positive value of "a" will satisfy (1.29a). We do not consider a < 0. Setting vo = -v + Zwm in (1.29b) gives (1.30) 2a sin v = v. For fixed a e (0.1) and any integer m. such that %E%' is non-integral, we find values of "a" which satis- fy (1.29). By assumption, the pairs (Vb'ao) which satisfy (1.29) correspond to the eigenvalues X(ao) = Re x(ao) + i Imx(ao) of (1.28) with Re x(ao) = O. For all pairs (Vb'ao) which satisfy (1.29) we first show that Re l’(a0) # 0. then, applying Theorem 5.1, concludes that (1.27) has non-zero periodic solutions. 6 TO show 33' Re 1| a=aO # O for all pairs (Vb'ao) which satisfy (1.29) we differentiate (1.28) with respect to a. we Obtain (1.31) Re l’(a = ) ao _ v0 Sin yo-v00 Sin “00 — r 2 2 . . 2 [(1HF300 COS vbG-ao cos vb) -ao(31n Wo"o Sln V00) ] 58 TABLE 3.3 o v a Re x’(a) .25 15.0796 12.8275 11.0795 .25 20.1062 10.5705 23.9027 25 40.2124 34.2067 29.545 .5 83.7758 48.3680 108.828 .5 71.2094 41.1128 92.5038 .5 58.6431 33.8576 76.1796 .5 46.0767 26.6024 ' 59.8554 .5 33.5103 19.3472 43.5312 .5 20.9440 12.0920 27.2070 .5 8.3776 4.8368 10.8828 .75 7.18078 4.59228 9.82478 .75 14.3616 7.36545 24.5026 .75 21.5423 24.8250 16.3570 .75 32.3135 20.6653 44.2115 At each pair tions in Table 3.3 show that Re x’(ao) # O. (vo.ao) which satisfy (1.29), the computa- The same remarks stated in Example 1 above applies here. Applying Theorem 5.1, (1.27) has non-zero periodic solutions. BIBLIOGRAPHY 10. 11. BIBLIOGRAPHY N.T.J. Bailey, The Mathematical Theory of Epidemics, Hafner, New Ybrk, 1957. S.N. Chow and J. Mallet-Paret, Fuller's index and glObal Hopf's bifurcation, to appear. K.L. CoOke and J.A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosciences, 16 (1973), 75-101. J.M. Greenberg, Periodic solutions to a pOpulation equation, to appear. J. Hale, Functional Differential Equations, Springer- Verlag, New York, 1971. E. Hopf, Abzweigung einer periodischen losung von einer stationaren losung eines differential systems, Ber. Math. Phys. K1. Sachs Akad. Wiss. Leipzig, 94 (1942). 1-22. F. H0ppensteadt and P. waltman, A prOblem in the theory of epidemics, II, Math. Bioscience, 12 (1971), 133-145. S. Lange, Differential Manifolds, 1972. J. Mosevich. A numerical method for approximating solutions to the functional equation arising in the epidemic model of Happensteadt and waltman, Math. Bioscience, 24 (1975), 333—344. M. Schechter, Principles of Functional Analysis, Academic Press, 1971. P. waltman, Deterministic threshold models in the theory of epidemics, Springer-Verlag, 1974. 59 "‘7r))))))))))1)))1“